We bought the house 3 years ago. The rose at the end of the picture below flowered just once in three years (this year). It grows like crazy, it is huge right now, almost bigger than the fence and I don't understand why it does not make flowers. What you see below is the only flowers we ever saw in three years.
Update 2: Yes the grafting theory makes sense but the solution you are suggesting is going to be difficult to implement as it is not clear what is the flowering part Below I marked the flowering part but it is difficult to say of which part of those old roots it is coming out. This rose used to be where the walkway you see in the initial picture was. It was relocated and I remember that my brother in law worked quite a lot to move it. It sat in the original position for about 1 year and a a half and we saw no flowers.